Is joint normality a necessary condition for the sum of normal random variables to be normal?

In comments following this answer of mine to a related question, Users ssdecontrol and Glen_b asked
whether joint normality of $X$ and $Y$ is necessary for asserting the
normality of the sum $X+Y$? That joint normality is sufficient is,
of course, well-known. This supplemental question was not addressed
there, and is perhaps worth considering in its own right.

Since joint normality implies marginal normality, I ask

Do there exist normal random variables $X$ and $Y$ such that
$X+Y$ is a normal random variable, but $X$ and $Y$ are not
jointly normal random variables?

If $X$ and $Y$ are not required to have normal distributions,
then it is easy to find such normal random variables. One example
can be found in my previous answer (link is given above).
I believe that the answer to the highlighted question above
is Yes, and have posted (what I think is) an example as an answer
to this question.


Let $U,V$ be iid $N(0,1)$.

Now transform $(U,V) \to (X,Y)$ as follows:

In the first quadrant (i.e. $U>0,V>0$) let $X=\max(U,V)$ and $Y = \min(U,V)$.

For the other quadrants, rotate this mapping about the origin.

The resulting bivariate distribution looks like (seen from above):

$\hspace{1.5 cm}$![enter image description here

— the purple represents regions with doubled probability and the white regions are ones with no probability. The black circles are contours of constant density (everywhere on the circle for $(U,V)$, but within each colored region for $(X,Y)$).

  1. By symmetry both $X$ and $Y$ are standard normal (looking down a vertical line or along a horizontal line there’s a purple point for every white one which we can regard as being flipped across the axis the horizontal or vertical line crosses)

  2. but $(X,Y)$ are clearly not bivariate normal, and

  3. $X+Y = U+V$ which is $\sim N(0,2)$ (equivalently, look along lines of constant $X+Y$ and see that we have symmetry similar to that we discussed in 1., but this time about the $Y=X$ line)

Source : Link , Question Author : Dilip Sarwate , Answer Author : Glen_b

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